Systems of Points with Coulomb Interactions

P. I. C. M. – 2018 Rio de Janeiro, Vol. 1 (935–978) SYSTEMS OF POINTS WITH COULOMB INTERACTIONS S S Abstract Large ensembles of points with Coulomb interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory, and give rise to a variety of questions pertaining to calculus of variations, Partial Differential Equations and probability. We will review these as well as “the mean-field limit” results that allow to derive effective models and equations describing the system at the macroscopic scale. We then explain how to analyze the next order beyond the mean-field limit, giving information on the system at the microscopic level. In the setting of statistical mechanics, this allows for instance to observe the effect of the temperature and to connect with crystallization questions. 1 General setups We are interested in large systems of points with Coulomb-type interactions, described through an energy of the form 1 N (1-1) HN (x1; : : : ; xN ) = X g(xi xj ) + N X V (xi ): 2 i j i=1 ¤ d Here the points xi belong to the Euclidean space R , although it is also interesting to consider points on manifolds. The interaction kernel g(x) is taken to be (1-2) (Log2 case) g(x) = log x ; in dimension d = 2; j j 1 (1-3) (Coul case) g(x) = ; in dimension d 3: x d 2 j j This is (up to a multiplicative constant) the Coulomb kernel in dimension d 2, i.e. the fundamental solution to the Laplace operator, solving (1-4) ∆g = cdı0 MSC2010: primary 60F05; secondary 60F15, 60K35, 60B30, 82B05, 82C22. Keywords: Coulomb gases, log gases, mean field limits, jellium, large deviations, point processes, crystallization, Gaussian Free Field. 935 936 SYLVIA SERFATY where ı0 is the Dirac mass at the origin and cd is an explicit constant depending only on the dimension. It is also interesting to broaden the study to the one-dimensional logarithmic case (1-5) (Log1 case) g(x) = log x ; in dimension d = 1; j j which is not Coulombic, and to more general Riesz interaction kernels of the form 1 (1-6) g(x) = s > 0: x s j j The one-dimensional Coulomb interaction with kernel x is also of interest, but we j j will not consider it as it has been extensively studied and understood, see Lenard [1961], Lenard [1963], and Kunz [1974]. Finally, we have included a possible external field or confining potential V , which is assumed to be regular enough and tending to fast enough at . The factor N in 1 1 front of V makes the total confinement energy of the same order as the total repulsion energy, effectively balancing them and confining the system to a subset of Rd of fixed size. Other choices of scaling would lead to systems of very large or very small size as N . ! 1 The Coulomb interaction and the Laplace operator are obviously extremely important and ubiquitous in physics as the fundamental interactions of nature (gravitational and electromagnetic) are Coulombic. Coulomb was a French engineer and physicist working in the late 18th century, who did a lot of work on applied mechanics (such as modeling friction and torsion) and is most famous for his theory of electrostatics and magnetism. He is the first one who postulated that the force exerted by charged particles is proportional to the inverse distance squared, which corresponds in dimension d = 3 to the gradient of the Coulomb potential energy g(x) as above. More precisely he wrote in Coulomb [1785] “ It follows therefore from these three tests, that the repul- sive force that the two balls [which were] electrified with the same kind of electricity exert on each other, follows the inverse proportion of the square of the distance.” He developed a method based on systematic use of mathematical calculus (with the help of suitable approximations) and mathematical modeling (in contemporary terms) to pre- dict physical behavior, systematically comparing the results with the measurements of the experiments he was designing and conducting himself. As such, he is considered as a pioneer of the “mathematization” of physics and in trusting fully the capacities of mathematics to transcribe physical phenomena Blondel and Wolff [2015]. Here we are more specifically focusing on Coulomb interactions between points, or in physics terms, discrete point charges. There are several mathematical problems that are interesting to study, all in the asymptotics of N : ! 1 (1) understand minimizers and possibly critical points of (1-1); (2) understand the statistical mechanics of systems with energy HN and inverse temperature ˇ > 0, governed by the so-called Gibbs measure 1 ˇHN (x1;:::;xN ) (1-7) dPN;ˇ (x1; : : : ; xN ) = e dx1 : : : dxN : ZN;ˇ SYSTEMS OF POINTS WITH COULOMB INTERACTIONS 937 Here, as postulated by statistical mechanics, PN;ˇ is the density of probability of observing the system in the configuration (x1; : : : ; xN ) if the number of particles is fixed to N and the inverse of the temperature is ˇ > 0. The constant ZN;ˇ is called the “partition function” in physics, it is the normalization constant that 1 makes PN;ˇ a probability measure, i.e. Z ˇHN (x1;:::;xN ) (1-8) ZN;ˇ = e dx1 : : : dxN ; (Rd)N where the inverse temperature ˇ = ˇN can be taken to depend on N , as there are several interesting scalings of ˇ relative to N ; (3) understand dynamic evolutions associated to (1-1), such as the gradient flow of HN given by the system of coupled ODEs 1 (1-9) x_i = i HN (x1; : : : ; xN ); N r conservative dynamics given by the systems of ODEs 1 (1-10) x_i = J i HN (x1; : : : ; xN ) N r where J is an antisymmetric matrix (for example a rotation by /2 in dimension 2), or the Hamiltonian dynamics given by Newton’s law 1 (1-11) x¨i = i HN (x1; : : : ; xN ); N r 1 (4) understand the previous dynamic evolutions with temperature ˇ in the form of an added noise (Langevin-type equations) such as 1 1 (1-12) dxi = i HN (x1; : : : ; xN )dt + pˇ dWi N r with Wi independent Brownian motions, or 1 1 (1-13) dxi = J i HN (x1; : : : ; xN )dt + pˇ dWi N r with J as above, or 1 1 (1-14) dxi = vi dt dvi = i HN (x1; : : : ; xN )dt + pˇ dWi : N r From a mathematical point of view, the study of such systems touches on the fields of analysis (Partial Differential Equations and calculus of variations, approximation theory) particularly for (1)-(3)-(4), probability (particularly for (2)-(4)), mathematical physics, and even geometry (when one considers such systems on manifolds or with 1One does not know how to explicitly compute the integrals (1-8) except in the particular case of (1-5) for specific V ’s where they are called Selberg integrals (cf. Mehta [2004] and Forrester [2010]) 938 SYLVIA SERFATY curved geometries). Some of the crystallization questions they lead to also overlap with number theory as we will see below. In the sequel we will mostly focus on the stationary settings (1) and (2), while men- tioning more briefly some results about (3) and (4), for which many questions remain open. Of course these various points are not unrelated, as for instance the Gibbs measure (1-7) can also be seen as an invariant measure for dynamics of the form (1-11) or (1-12). The plan of the discussion is as follows: in the next section we review various motivations for studying such questions, whether from physics or within mathematics. In Section 3 we turn to the so-called “mean-field” or leading order description of systems (1) to (4) and review the standard questions and known results. We emphasize that this part can be extended to general interaction kernels g, starting with regular (smooth) interactions which are in fact the easiest to treat. In Section 4, we discuss questions that can be asked and results that can be obtained at the next order level of expansion of the energy. This has only been tackled for problems (1) and (2), and the specificity of the Coulomb interaction becomes important then. 2 Motivations It is in fact impossible to list all possible topics in which such systems arise, as they are really numerous. We will attempt to give a short, necessarily biased, list of examples, with possible pointers to the relevant literature. 2.1 Vortices in condensed matter physics and fluids. In superconductors with applied magnetic fields, and in rotating superfluids and Bose–Einstein condensates, one observes the occurrence of quantized “vortices” (which are local point defects of superconductivity or superfluidity, surrounded by a current loop). The vortices repel each other, while being confined together by the effect of the magnetic field or rotation, and the result of the competition between these two effects is that, as predicted by Abrikosov [1957], they arrange themselves in a particular triangular lattice pattern, called Abrikosov lattice, cf. Fig. 12. Superconductors and superfluids are modelled by the celebrated Ginzburg–Landau energy Landau and Ginzburg [1965], which in simpli- fied form 3 can be written (1 2)2 (2-1) Z 2 + j j jr j 2"2 where is a complex-valued unknown function (the “order parameter” in physics) and " is a small parameter, and gives rise to the associated Ginzburg–Landau equation 1 (2-2) ∆ + (1 2) = 0 "2 j j 2For more pictures, see www.fys.uio.no/super/vortex/ 3The complete form for superconductivity contains a gauge-field, but we omit it here for simplicity.

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